Research

In the summers of 2018, 2019, and 2020, I had the privilege of participating in the University of Chicago REU.

2018

My project in 2018 was called “Algebraic and Analytic Properties of Arithmetic Functions”. I was interested in the algebraic properties of the set of functions \(\mathbb{N} \to \mathbb{N}\). When equipped with the operations of pointwise addition and Dirichlet convolution, this set becomes an integral domain, under which the theorem of Mobius inversion has a simplified, more natural statement. I proved original properties of this domain, including a conditional analog of fundamental theorem of algebra, and analyzed the growth properties of some arithmetic functions.
The writeup can be found here.

2019

My project in 2019 was called "The Game of Hex: A Study in Graph Theory and Algebraic Topology". Hex is a game invented in the 1940s with the non-trivial property that no completed game can end in a tie. I studied this Hex theorem, at its graph-theoretic roots, as well as its topological consequences, and presented an original generalization to the torus which preserves this property.
The writeup can be found here.

2020

In 2020, my project was called "Enlargements and the Non-Standard Perspective". I studied the theory of enlarging models, as it pertained to Abraham Robinson’s nonstandard analysis and Ramsey theory. I summarized the numerous simplifications both of definitions and of proofs which this perspective affords, while assuming only minimal background in mathematical logic and model theory.
The writeup can be found here.